Past

Note: Change of time and (for Logic) place! Joint with Number Theory (double header)

Quivers are directed graphs which can be thought of as "space" in noncommutative geometry. In this talk, we will try to establish a link between noncommutative geometry and its commutative counterpart. We will show how one can construct (differential graded) quivers which are "equivalent" (in the sense of derived category of representations) to vector bundles on smooth varieties.

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

Much of the mathematical modelling of urban systems revolves around the use spatial interaction models, derived from information theory and entropy-maximisation techniques and embedded in dynamic difference equations. When framed in the context of a retail system, the

dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be analysed analytically. In this contribution, we present some analysis of the continuous retail model and corresponding discrete version, which yields insights into the effect of space on the system, and an understanding of why certain retail centers are more successful than others. This class of models turns out to have wide reaching applications: from trade and migration flows to the spread of riots and the prediction of archeological sites of interest, examples of which we explore in more detail during the talk.

The goal of this lecture is describing recent joint work with Henri Darmon, in which we construct an Euler system of twisted Gross-Kudla diagonal cycles that allows us to prove, among other results, the following statement (under a mild non-vanishing hypothesis that we shall make explicit):

Let $E/\mathbb{Q}$ be an elliptic curve and $K=\mathbb{Q}(\sqrt{D})$ be a real quadratic field. Let $\psi: \mathrm{Gal}(H/K) \rightarrow \mathbb{C}^\times$ be an anticyclotomic character. If $L(E/K,\psi,1)\ne 0$ then the $\psi$-isotypic component of the Mordell-Weil group $E(H)$ is trivial.

Such a result was known to be a consequence of the conjectures on Stark-Heegner points that Darmon formulated at the turn of the century. While these conjectures still remain highly open, our proof is unconditional and makes no use of this theory.

Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using ``local’’ techniques, such as RBF generated finite differences (RBF-FD).

In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of ``scattered’’ nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.

This talk represents joint work with Edward Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).

A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.

For a finitely presented group, the Word Problem asks for an algorithm

which declares whether or not words on the generators represent the

identity. The Dehn function is the time-complexity of a direct attack

on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast

growing (Ackermannian) Dehn functions. I will explain why,

nevertheless, there are efficient (polynomial time) solutions to the

Word Problems of these groups. The main innovation is a means of

computing efficiently with compressed forms of enormous integers.

This is joint work with Will Dison and Eduard Einstein.

Quantifying the uncertainty in computational simulations is one of the central challenges confronting the field of computational science and engineering today. The uncertainty quantification of inverse problems is neatly addressed in the Bayesian framework, where instead of seeking one unique minimiser of a regularised misfit functional, the entire posterior probability distribution is to be characterised. In this talk I review the deep connection between deterministic PDE-constrained optimisation techniques and Bayesian inference for inverse problems, discuss some recent advances made in the Bayesian viewpoint by adapting deterministic techniques, and mention directions for future research.

Composite dilation wavelets are affine systems which extend the notion of wavelets by incorporating a second set of dilations.  The addition of a second set of dilations allows the composite system to capture directional information in addition to time and frequency information.  We classify admissible dilation groups at two extremes: frequency localization through minimally supported frequency composite dilation wavelets and time localization through crystallographic Haar-type composite dilation wavelets. 

The instanton corrections to the hyperkähler metric on moduli spaces of meromorphic flat SL(2,C)-connections on a Riemann surface with prescribed singularities have recently been studied by Gaiotto, Moore and Neitzke. The instantons are given by certain special trajectories of the meromorphic quadratic differentials which form the base of Hitchin's integrable system structure on the moduli space. Bridgeland and Smith interpret such quadratic differentials as defining stability conditions on an associated 3-Calabi-Yau triangulated category whose stable objects correspond to these special trajectories.

The smallest non-trivial examples are provided by the moduli spaces of quaternionic dimension one. In these cases it is possible to study explicitly the periods of the Seiberg-Witten differential on the fibres of the Hitchin system which define the central charge of the stability condition and lift the period map to the space of stability conditions. This provides in particular a new categorical perspective on the original Seiberg-Witten gauge theories.

The talk is based on my paper with E. Beggs appearing in Class. Quantum

Gravity.

Working within a bimodule approach to noncommutative geometry, we show that

even a small amount of noncommutativity drastically constrains the moduli

space of

noncommutative metrics. In particular, the algebra [x,t]=x is forced to have

a geometry

corresponding to a gravitational source at x=0 so strong that even light

cannot

escape. This provides a non-trivial example of noncommutative Riemannian

geometry

and also serves as an introduction to some general results.

One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,

transonic shock can be represented as a monotone function of exit pressure.

From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system

when exit pressure is prescribed in a proper range.

In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,

which is formulated as a free boundary problem with mixed type PDE system.

This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao

In 1997 V. Bentkus and F. Götze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.

The'signature', from the theory of differential equations driven by rough paths,
provides a very efficient way of characterizing curves. From a machine learning
perspective, the elements of the signature can be used as a set of features for
consumption by a classification algorithm.

Using datasets of letters, digits, Indian characters and Chinese characters, we
see that this improves the accuracy of online character recognition---that is
the task of reading characters represented as a collection of pen strokes.

A "bordism representation" (*) is a representation of the abstract

structure formed by manifolds and bordisms between them, and hence of

fundamental interest in topology. I will give an overview of joint work

establishing a simple generators-and-relations presentation of the

3-dimensional oriented bordism bicategory, and also its "signature" central

extension. A representation of this bicategory corresponds in a 2-1 fashion

to a modular category, which must be anomaly-free in the oriented case. J/w

Chris Douglas, Chris Schommer-Pries, Jamie Vicary.

(*) These are also known as "topological quantum field theories".

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

Jakobshavn Isbrae and many other fast flowing outlet glaciers of present

and past ice sheets lie in deep troughs which often have several

overdeepened sections. To make their fast flow possible their bed needs

to be slippery which in turn means high basal water pressures. I will

present a model of subglacial water flow and its application to

Jakobshavn. I find that, somewhat surprisingly, the reason for

Jakobshavn's fast flow might be the pressure dependence of the melting

point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.

We consider the pricing of American put options in a model-independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American Put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process. (Joint with Christoph Hoeggerl).

A pseudofinite group is an infinite model of the theory of finite groups. I will discuss what can be said about pseudofinite groups under various tameness assumptions on the theory (e.g. NIP, supersimplicity), structural results on pseudofinite permutation groups, and connections to word maps and generalisations.

Ricci solitons were introduced by Richard Hamilton in the 80's and they are a generalization of the better know Einstein metrics. During this talk we will define the notion of Ricci soliton and I will try to convince you that these metrics arise "naturally" in a number of different settings. I will also present various examples and talk a bit about some symmetry properties that Ricci solitons have.

Note: This talk is meant to be introductory and no prior knowledge about Einstein metrics will be assumed (or necessary).

After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.

This is joint work with É. Fouvry and Ph. Michel.

Many real-life complex systems arise as a network of simple interconnected individual agents. A central question is to determine how network topology and individual agent dynamics combine to create the global dynamics.

In this talk we focus on the case of continuous-time random walks on networks, with a waiting time of the walker on each node assuming arbitrary probability distributions. Such random walks are useful to model diffusion processes over complex temporal networks representing human interactions, often characterized by non-Poissonian contact patterns.

We find that the mixing time of the random walker, i.e. the relaxation time for the process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show

theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data.

These results are available in arXiv:1309.4155

TBA

The fundamental mechanisms of microorganism motility have been extensively studied in the past. Most previous work focused on cell locomotion in simple (Newtonian) fluids.
However, in many cases of biological importance (including mammalian reproduction and bacterial infections), the fluids that surround the organisms are strongly non-Newtonian (so-called complex fluids), either because they have shear-dependent viscosities, or because they display an elastic response. These non-Newtonian effects challenge the most fundamental intuition in fluid mechanics, resulting in our incapacity to predict its implications in biological cell locomotion. In this talk, our on-going experimental investigation to quantify the effect of non-Newtonian behavior on the locomotion and fluid transport of microorganisms will be described. Several types of magnetic micro-robots were designed and built. These devices were actuated to swim or move in a variety of fluids : Newtonian, elastic with constant viscosity (Boger fluids) or inelastic with shear-thinning viscosity. We have found that, depending on the details of locomotion, the swimming performance can either be increased, decreased or remain unaffected by the non Newtonian nature of the liquid. Some key elements to understand the general effect of viscoelasticity and shear-thinning viscosity of the motility of microorganisms will be discussed.

In this talk we will show the existence  of a regular "small" weak solution to the flow of the higher dimensional H-systems with initial-boundary conditions. We also analyze its time asymptotic bahavior and we give a stability result.

This talk aims to illustrate how graphical calculus can be used to reason about Hopf algebras and their modules. The talk will be aimed at a general audience requiring no previous knowledge of the topic.

A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient.

In 1853, Steiner posed the natural generalisation of the question: given integers q and r, for which n is it possible to choose a collection Q of q-element subsets of an n-element set X such that any r elements of X are contained in exactly one of the sets in Q? There are some natural necessary divisibility conditions generalising the necessary conditions for Steiner Triple Systems. The Existence Conjecture states that for all but finitely many n these divisibility conditions are also sufficient for the existence of general Steiner systems (and more generally designs).

We prove the Existence Conjecture, and more generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.

Convolution is widely-used and fundamental mathematical operation
in signal processing, statistics, and PDE theory.

Unfortunately the CONV() method in Chebfun for convolving two chebfun 
objects has long been one of the most disappointingly slow features of 
the project. In this talk we will present a new algorithm, which shows 
performance gains on the order of a factor 100.

The key components of the new algorithm are:
* a convolution theorem for Legendre polynomials 
* recurrence relations satisfied by spherical Bessel functions
* recent developments in fast Chebyshev-Legendre transforms [1]

Time-permitting, we shall end with an application from statistics,
using the fact that the probability distribution of the sum of two 
independent random variables is the convolution of their individual 
distributions.

[1] N. Hale and A. Townsend, "A fast, simple, and stable Chebyshev-
Legendre transform using an asymptotic formula”, SISC (to appear).

The accurate and efficient solution of linear systems $Ax=b$ is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.  This work is joint with Jacek Gondzio.

The finance literature documents a relation between labor income and

the cross-section of stock returns. One possible explanation for this

is the hedging decisions of investors with relative wealth concerns.

This implies a negative risk premium associated with stock returns

correlated with local undiversifiable wealth, since investors are

willing to pay more for stocks that help their hedging goals. We find

evidence that is consistent with these regularities. In addition, we

show that the effect varies across geographic areas depending on the

size and variability of undiversifiable wealth, proxied by labor income.

This is another opportunity to hear the 2013 LMS Presidential Address:

Abstract: The idea of space is central to the way we think.  It is the technology we have evolved for interpreting our experience of the world.  But space is presumably a human creation, and even inside mathematics it plays a variety of different roles, some modelling our intuition very closely and some seeming almost magical.  I shall point out how the homotopy category in particular breaks away from its own roots.  Then I shall describe how quantum theory leads us beyond the well-established notion of a topological space into the realm of noncommutative geometry.  One might think that noncommutative spaces are not very space-like, and yet it is noncommutativity that makes the world look as it does to us, as a collection of point particles.

In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.

Liquid Crystals (LC), anisotropic fluids that combine many tensor properties of crystalline solids with the fluidity of liquids, have long been providing major challenges to theorists and molecular modelers. In the classical textbook picture a molecule giving rise to LC phases is represented by a uniaxial rod endowed with repulsive (Onsager) or attractive (Maier-Saupe) interactions or possibly with a combination of the two (van der Waals picture) [1]. While these models have proved able to reproduce at least qualitatively the most common LC phase, the nematic one, and its phase transition to a normal isotropic fluid, they have not been able to deal with quantitative aspects (e.g. the orientational order at the transition) and more seriously, with the variety of novel LC phases and of sophisticated experiments offering increasing detailed observations at the nanoscale. Classical Monte Carlo and molecular dynamics computer simulations that have been successfully used for some time on simple lattice or off-lattice generic models [2-5] have started to offer unprecedented, atomistic level, details of the molecular organization of LC in the bulk and close to surfaces [6,7]. In particular, atomistic simulations are now starting to offer predictive power, opening the possibility of closing the gap between molecular structure and phase organizations. The availability of detailed data from these virtual experiments requires to generalize LC models inserting molecular features like deviation from uniaxiality or rigidity, the inclusion of partial charges etc. Such more detailed descriptions should reflect also in the link between molecular and continuum theories, already developed for the simplest models [8,9], possibly opening the way to a molecular identification of the material and temperature dependent coefficients in Landau-deGennes type free energy functionals.

[1] see, e.g., G. R. Luckhurst and G. W. Gray, eds., The Molecular Physics of Liquid Crystals (Academic Press,, 1979).

[2] P. Pasini and C. Zannoni, eds., Advances in the computer simulations of liquid crystals (Kluwer, 1998)

[3] O. D. Lavrentovich, P. Pasini, C. Zannoni and S. Zumer, eds. Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, (Kluwer, Dordrecht , 2001).

[4] C. Zannoni, Molecular design and computer simulations of novel mesophases, J. Mat. Chem. 11, 2637 (2001).

[5] R.Berardi, L.Muccioli, S.Orlandi, M.Ricci, C.Zannoni, Computer simulations of biaxial nematics, J. Phys. Cond. Matter 20, 1 (2008).

[6] G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni , Towards “in silico” liquid crystals. Realistic Transition temperatures and physical properties for n-cyanobiphenyls via molecular dynamics simulations, ChemPhysChem 10, 125 (2009).

[7] O. Roscioni, L. Muccioli, R. Della Valle, A. Pizzirusso, M. Ricci and C. Zannoni, Predicting the anchoring of liquid crystals at a solid surface: 5-cyanobiphenyl on cristobalite and glassy silica surfaces of increasing roughness, Langmuir 29, 8950 (2013).

[8] 1. J. Katriel, G. F. Kventsel, G. R. Luckhurst and T. J. Sluckin, Free-energies in the Landau and Molecular-field approaches, Liq. Cryst. 1, 337 (1986).

[9] J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a Continuum Theory, Mol. Cryst. Liq. Cryst. 525, 1 (2010).

Computational structures---from simple objects like bits and qubits,

to complex procedures like encryption and quantum teleportation---can

be defined using algebraic structures in a symmetric monoidal

2-category. I will show how this works, and demonstrate how the

representation theory of these structures allows us to recover the

ordinary computational concepts. The structures are topological in

nature, reflecting a close relationship between topology and

computation, and allowing a completely graphical proof style that

makes computations easy to understand. The formalism also gives

insight into contentious issues in the foundations of quantum

computing. No prior knowledge of computer science or category theory

will be required to understand this talk.

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties.  Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown.  I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto.  In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.